Absolute Value Function: Evaluate, Domain, Range, And Graph

Hey guys! Today, we're diving deep into the fascinating world of absolute value functions. We'll be tackling a problem that involves evaluating an absolute value function at specific points, determining its domain and range, and even sketching its graph. So, buckle up and let's get started!

Evaluating the Absolute Value Function

Let's kick things off by evaluating the absolute value function f(x) = |x - 3| at the given points. Remember, the absolute value of a number is its distance from zero, so it's always non-negative. This means we need to consider two cases: when the expression inside the absolute value is positive or zero, and when it's negative.

Evaluating at x = -2

First, we'll evaluate f(x) at x = -2. We substitute -2 for x in the function:

f(-2) = |-2 - 3| = |-5| = 5

So, when x = -2, the value of the function is 5. This makes sense because the absolute value ensures that the result is always positive, representing the distance from zero. The calculation is straightforward: we subtract 3 from -2, which gives us -5, and then take the absolute value of -5, which is 5.

Evaluating at x = 0

Next up, we'll evaluate f(x) at x = 0:

f(0) = |0 - 3| = |-3| = 3

When x = 0, the function value is 3. Again, the absolute value ensures a positive outcome. We subtract 3 from 0, resulting in -3, and the absolute value of -3 is 3. This illustrates how the absolute value function transforms any negative result into its positive counterpart, maintaining the distance aspect.

Evaluating at x = -5/4

Now, let's tackle a slightly trickier one: x = -5/4. Don't worry, we'll break it down step by step:

f(-5/4) = |-5/4 - 3| = |-5/4 - 12/4| = |-17/4| = 17/4

So, when x = -5/4, the function value is 17/4. We first convert 3 to a fraction with a denominator of 4 to combine it with -5/4, resulting in -12/4. Subtracting 12/4 from -5/4 gives us -17/4, and the absolute value of -17/4 is 17/4. This demonstrates how the function handles fractions and maintains the principle of absolute value.

Evaluating at x = 5

Finally, let's evaluate f(x) at x = 5:

f(5) = |5 - 3| = |2| = 2

When x = 5, the function value is 2. This is perhaps the most straightforward calculation: subtracting 3 from 5 gives us 2, and the absolute value of 2 is simply 2. This reinforces the concept that the absolute value of a positive number is the number itself.

In summary, evaluating the absolute value function at different points involves substituting the given value into the function and simplifying. The absolute value ensures that the final result is always non-negative, representing the distance from zero. Whether the input is negative, zero, or positive, the absolute value function transforms it into its non-negative equivalent.

Domain and Range of the Absolute Value Function

Next, let's discuss the domain and range of the absolute value function f(x) = |x - 3|. Understanding the domain and range is crucial for comprehending the function's behavior and its possible inputs and outputs.

Domain of f(x) = |x - 3|

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the absolute value function f(x) = |x - 3|, there are no restrictions on the values we can plug in for x. We can input any real number into the function, and it will produce a valid output. This is because the absolute value is defined for all real numbers. Therefore, the domain of f(x) is all real numbers.

In mathematical notation, we can express the domain as:

Domain: (-∞, ∞)

This notation indicates that the domain includes all real numbers from negative infinity to positive infinity. There are no values of x that would make the function undefined, such as division by zero or taking the square root of a negative number. The absolute value function is robust and can handle any real number input.

Range of f(x) = |x - 3|

The range of a function is the set of all possible output values (y-values or f(x) values) that the function can produce. Since the absolute value function always returns a non-negative value, the range of f(x) = |x - 3| will be all non-negative real numbers. The minimum value of the function occurs when the expression inside the absolute value is equal to zero. In this case, that happens when x = 3:

f(3) = |3 - 3| = |0| = 0

So, the minimum value of the function is 0. As x moves away from 3 in either direction (positive or negative), the value of |x - 3| increases. Therefore, the function can take on any non-negative value.

In mathematical notation, we express the range as:

Range: [0, ∞)

This notation indicates that the range includes all real numbers greater than or equal to 0, extending to positive infinity. The square bracket on the 0 indicates that 0 is included in the range, while the parenthesis on ∞ indicates that infinity is not a specific number and is not included.

In summary, the domain of f(x) = |x - 3| is all real numbers, meaning we can input any real number into the function. The range of f(x) = |x - 3| is all non-negative real numbers, meaning the output of the function will always be 0 or a positive number. Understanding the domain and range helps us visualize the function's behavior and the set of possible values it can take.

Sketching the Graph of the Absolute Value Function

Now, let's move on to sketching the graph of the absolute value function f(x) = |x - 3|. Visualizing the graph is a great way to understand the function's behavior and characteristics. We'll use the information we've already gathered about the function's domain, range, and some key points to create an accurate sketch.

Key Features of the Graph

Before we start plotting points, let's consider the key features of the graph of f(x) = |x - 3|:

1. V-Shape: Absolute value functions have a characteristic V-shape. The vertex (the point where the V changes direction) is a crucial point to identify.
2. Vertex: The vertex of the graph occurs where the expression inside the absolute value is equal to zero. In this case, x - 3 = 0, so the vertex occurs at x = 3. The corresponding y-value is f(3) = |3 - 3| = 0. Thus, the vertex is at the point (3, 0).
3. Symmetry: The graph is symmetric about the vertical line that passes through the vertex. This means that the left and right sides of the V-shape are mirror images of each other.
4. Domain and Range: As we discussed earlier, the domain is all real numbers, and the range is [0, ∞). This tells us that the graph extends infinitely to the left and right, and it is always above or on the x-axis.

Plotting Key Points

To sketch the graph, we'll plot a few key points. We already know the vertex is at (3, 0). Let's find a few other points by evaluating the function at different x-values:

• When x = 2: f(2) = |2 - 3| = |-1| = 1. So, we have the point (2, 1).
• When x = 4: f(4) = |4 - 3| = |1| = 1. So, we have the point (4, 1).
• When x = 1: f(1) = |1 - 3| = |-2| = 2. So, we have the point (1, 2).
• When x = 5: f(5) = |5 - 3| = |2| = 2. So, we have the point (5, 2).

Sketching the Graph

Now, we can plot these points on a coordinate plane: (3, 0), (2, 1), (4, 1), (1, 2), and (5, 2). We can see the V-shape forming. Draw a straight line connecting the points to the left of the vertex, and another straight line connecting the points to the right of the vertex. The lines should extend infinitely in both directions, representing the domain of the function.

The graph of f(x) = |x - 3| will look like a V-shape with the vertex at (3, 0). The left side of the V extends from (3, 0) through (2, 1) and (1, 2), and the right side extends from (3, 0) through (4, 1) and (5, 2). The graph is symmetric about the vertical line x = 3.

Understanding the Transformation

It's worth noting that the graph of f(x) = |x - 3| is a transformation of the basic absolute value function g(x) = |x|. The transformation is a horizontal shift of 3 units to the right. The graph of g(x) = |x| has its vertex at (0, 0), while the graph of f(x) = |x - 3| has its vertex at (3, 0). This shift is due to the -3 inside the absolute value.

In summary, sketching the graph of f(x) = |x - 3| involves identifying the V-shape, finding the vertex, plotting key points, and drawing lines to connect the points. Understanding the domain, range, and transformations helps in creating an accurate sketch. Visualizing the graph provides a clear understanding of the function's behavior and its properties.

Conclusion

Alright, guys, we've covered a lot about the absolute value function f(x) = |x - 3|. We evaluated it at different points, determined its domain and range, and sketched its graph. Hopefully, you now have a solid understanding of how these functions work. Keep practicing, and you'll become absolute value function masters in no time! Remember, math is all about understanding the concepts and applying them. Until next time, keep learning and exploring! This problem helps us understand the fundamentals of absolute value functions, which are crucial in various areas of mathematics and real-world applications.